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Questions tagged [quadratics]

Questions about quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$.

Questions about quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$.

The root of $y=ax^2+bx+c$ can be solved by the formula $$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

5400 questions
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Finding line of symmetry and x-intercepts in simple quadratic

Ok so in a non calculator (mental) practice question. I've been given a quadratic equation in the form $y = ax^2 + bx + c$: $y= 2 + 1.75x - 0.25x^2$ After rearranging and changing the decimals to fractions I end up with: $y = (-1/4)x^2 + (7/4)x…
A.Mahony
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Quadratic roots greater than 1 for $1-\alpha_1 x - \alpha_2 x^2$

I am trying to prove the conditions that need to be valid for the below quadratic to have roots outside the unit circle. $1-\alpha_1 x - \alpha_2 x^2$ and the conditions that need to hold are: $|\alpha_2| < 1 $ ; $\alpha_2 + \alpha_1 < 1 $ …
yudi
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Question on quadratic inequations

Given an equation $ax^2+bx+c$ $>$ or $<$ 0. Note that $a$ is kept positive. If the discriminant $(\sqrt{b^2-4ac})$ of the quadratic equation $< 0$, then the solution either applies for all real $x$ or no real $x$ depending on the inequality sign…
Mathejunior
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To find the least possible value of $c$

suppose $f(x) = -x^2 + bx + 1$ and and $g(x) = x^2 + 2x + c$ are such that $max \quad f(x)\le min\quad g(x)$ as $x$ varies over the set of real numbers. The least possible value of $c$ what I tried is to the maximum value of $f(x)$ and minimum value…
biswa prakash mishra
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Factorizing/solving for x in simple quadratic

I have a quadratic equation: $$6x^2 - 19x + 10 = 0$$ I don't understand how to factorize it and thus solve for $x$ because of the $6$ out the front. If someone could show me in steps how to achieve the answer of $x=\frac23$, $x=\frac52$, it would be…
A.Mahony
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Simplification of equation

How to get alternative form from equation 1) $$ 1) -a^2 + a + b^2 -b $$ to equation 2) $$ 2) (a-b)(a+b-1)$$
A.D.
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Symmetric quadratic, positive root less than one

I have this "symmetric" quadratic equation: \begin{align*} a(1-x)^2+b(1-x)x+c x^2 = 0 \end{align*} and I am trying to impose conditions on $a,b,c$ such that there is at least one root $0
folouer of spiros
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What is the condition of the coefficients of $ax^2+bx+c$ for both the roots being equal and imaginary.

What is the relation between the coefficients of $ax^2+bx+c$ for both the roots to be equal and imaginary. See we know that $b^2-4ac=0$ has roots equal and real. This may be very elementary but its is bothering me for a few days.
Pole_Star
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Quadratic equation variables

Suppose that $3+2\sqrt2$ solves $x^2-6x+a=0$. Find the value of $a$. a) $1$ b) $3$ c) $5$ d) $6$ option e) is cut out of the picture and it is $8$. I want a step by step explanation on how to solve this.
gommb
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Quadratic equation problems

Find the range of values of h if $g(x)=3 x²+2 x+2 h$ is always positive. Should I use $b²-4 a c>0$ to get the correct answer?
Tai Sheng Hong
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Satisfying a quadratic equation

I'm having a bit of trouble with understanding how to satisfying quadratic equations with more than one variable. Could you help me with this question please? If we're given these two conditions: $$a/b = b/(a-b)$$ $$x = a/b$$ How do we show that $x$…
Somnifirim
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Quadratic Problem sum imvolving financials

A man buys some bottle for $ \text{300} $ and when offloading he breaks $10$. He sells the remaining with a mark up of $4$ and makes $100$ profit on the entire transaction. How many bottles did he buy? I know the cost of the bottle is $300/x$ but…
Yusuf Essop
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Proof for no positive real roots

Prove that $x^6$ + $x^4$ +$x^2$+ $x$ + $3$ has no positive real roots. My attempt: I couldn't really think of a way to do this. I have looked at a similar problem but it had a cubic equation where you assign variables to the roots and use them to…
Amit Hegde
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Finding the other root of the quadratic equation in specific form

This question has been solved. However, I notice that both the solutions (including the comment) are verifying the fact that $4\alpha^3-3\alpha$ is the other root if $\alpha$ is one. What if I rephrase the question as:- If $\alpha$ is a root of the…
Mick
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The quickest way to solve a quadratic equation with imaginary solution

I had an exam question for solving an equation with imaginary roots: $$ m^2-3m+6=0 $$ Obviously this yields imaginary roots. What's the quickest way to find the solution?
Nipesh Kc
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